Properties

Label 2-2151-1.1-c1-0-71
Degree $2$
Conductor $2151$
Sign $-1$
Analytic cond. $17.1758$
Root an. cond. $4.14437$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.12·2-s + 2.49·4-s + 2.47·5-s − 2.45·7-s − 1.05·8-s − 5.23·10-s + 4.01·11-s + 1.62·13-s + 5.20·14-s − 2.76·16-s − 4.89·17-s − 6.28·19-s + 6.16·20-s − 8.50·22-s + 3.83·23-s + 1.10·25-s − 3.44·26-s − 6.12·28-s + 0.357·29-s − 8.44·31-s + 7.96·32-s + 10.3·34-s − 6.06·35-s − 1.81·37-s + 13.3·38-s − 2.59·40-s − 10.7·41-s + ⋯
L(s)  = 1  − 1.49·2-s + 1.24·4-s + 1.10·5-s − 0.927·7-s − 0.371·8-s − 1.65·10-s + 1.20·11-s + 0.450·13-s + 1.38·14-s − 0.690·16-s − 1.18·17-s − 1.44·19-s + 1.37·20-s − 1.81·22-s + 0.798·23-s + 0.220·25-s − 0.676·26-s − 1.15·28-s + 0.0663·29-s − 1.51·31-s + 1.40·32-s + 1.78·34-s − 1.02·35-s − 0.298·37-s + 2.16·38-s − 0.410·40-s − 1.67·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2151\)    =    \(3^{2} \cdot 239\)
Sign: $-1$
Analytic conductor: \(17.1758\)
Root analytic conductor: \(4.14437\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2151} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 2151,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
239 \( 1 - T \)
good2 \( 1 + 2.12T + 2T^{2} \)
5 \( 1 - 2.47T + 5T^{2} \)
7 \( 1 + 2.45T + 7T^{2} \)
11 \( 1 - 4.01T + 11T^{2} \)
13 \( 1 - 1.62T + 13T^{2} \)
17 \( 1 + 4.89T + 17T^{2} \)
19 \( 1 + 6.28T + 19T^{2} \)
23 \( 1 - 3.83T + 23T^{2} \)
29 \( 1 - 0.357T + 29T^{2} \)
31 \( 1 + 8.44T + 31T^{2} \)
37 \( 1 + 1.81T + 37T^{2} \)
41 \( 1 + 10.7T + 41T^{2} \)
43 \( 1 - 9.09T + 43T^{2} \)
47 \( 1 + 11.0T + 47T^{2} \)
53 \( 1 + 5.78T + 53T^{2} \)
59 \( 1 - 1.43T + 59T^{2} \)
61 \( 1 - 2.21T + 61T^{2} \)
67 \( 1 + 14.2T + 67T^{2} \)
71 \( 1 - 4.12T + 71T^{2} \)
73 \( 1 - 2.78T + 73T^{2} \)
79 \( 1 - 7.84T + 79T^{2} \)
83 \( 1 - 9.91T + 83T^{2} \)
89 \( 1 + 0.978T + 89T^{2} \)
97 \( 1 + 16.8T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.971522014725962878691639390974, −8.282033800799448440018512941374, −6.97781655643342866845423055451, −6.60614557595836670900855575796, −5.99320254414350147469878186562, −4.65054950404136265284566193363, −3.52591331309711736655606884272, −2.20051754880824364210673340836, −1.51044915052966091444512786167, 0, 1.51044915052966091444512786167, 2.20051754880824364210673340836, 3.52591331309711736655606884272, 4.65054950404136265284566193363, 5.99320254414350147469878186562, 6.60614557595836670900855575796, 6.97781655643342866845423055451, 8.282033800799448440018512941374, 8.971522014725962878691639390974

Graph of the $Z$-function along the critical line